2. Dynamic characteristic modeling and simulation of ball screw feed system

#### 2.1. Lumped mass model of ball screw feed system

A typical ball screw feed system consists of a servomotor, coupling, ball screw, work table, and base (Figure 1). The ball screw is supported by two sets of bearing, which are fixed to the base. The servomotor torque is transmitted through a coupling onto the ball screw shaft to drive the work table. The linear guideway constrains the movement of the work table in an axial direction. The base is fixed on the machine bed or placed on the ground. The transformation from the rotational movement of the screw shaft into the linear motion of the work table is realized by the ball screw system with its transmission ratio i, which is defined as the distance of travel h during one revolution of the shaft as the following:

$$i = \frac{h}{2\pi} \tag{1}$$

Low-order modes are the main factors affecting the dynamic characteristics of the ball screw feed drive system of machine tools. Typically, the first axial and rotational modes of the ball screw show a dominant influence on the overall dynamics, while the relevance of higher-order modes for most technical applications is rather small [8].

The lumped mass model can reasonably reduce the number of degrees of freedom (DOF) of the simulation model while preserving the low-order modes of the system to simplify calculations. Figure 2 shows the lumped mass model of a ball screw feed drive system. The influence of the shaft on the rotational mode and axial mode of the drive system is explicitly included into the lumped mass model here. Therefore, the shaft is separated into two different branches, an axial branch and a rotational branch, while the coupling once more is realized using constrained equations. Since all components are expressed by discrete springs and dampers, the rigidity values of shaft, coupling, and bearing are combined to an overall axial Kax and rotational value Krot.

way to solve this problem is to generate a smoother trajectory. For this purpose, numbers of trajectory algorithms are found out, and the frequency contents of the trajectory orders are discussed and compared [5, 6]. The vibration caused by the trajectory is difficult to analyze on hardware because of the coupling factor of variety excitation sources. All these researches need a simulation method to help the researchers or engineers study or optimize the design and

Finite element model of ball screw feed drive system can predict the accurate dynamic characteristics. However, it is difficult to integrate with the simulation model of servo control system. Lumped parameter model of ball screw feed drive system can simplify the simulation model by reducing the number of degrees of freedom (DOF) of the whole system. More importantly, it can easily integrate with the simulation model of the servo control system. A reasonable simplification of the lumped parameter model is the key to accurately predict the vibration of feed drive system [7–9]. In this chapter an electromechanical co-simulation method for ball screw feed drive system was established, which can be used to study the dynamic characteristics and vibration behavior of the feed drive system. An optimized dynamic modeling and simulation method of a ball screw feed drive based on the lumped mass model was firstly presented, and the optimized calculation method of the equivalent parameters was given. Then, a model of servo control system was built up, and based on it, the electromechanical co-simulation of ball screw feed drive system was established. Finally, a simulative and experimental test is conducted based on a ball screw feed drive system test bench. The result shows that electromechanical co-

simulation of ball screw feed drive system could achieve a very good predictability.

2. Dynamic characteristic modeling and simulation of ball screw feed

A typical ball screw feed system consists of a servomotor, coupling, ball screw, work table, and base (Figure 1). The ball screw is supported by two sets of bearing, which are fixed to the base. The servomotor torque is transmitted through a coupling onto the ball screw shaft to drive the work table. The linear guideway constrains the movement of the work table in an axial direction. The base is fixed on the machine bed or placed on the ground. The transformation from the rotational movement of the screw shaft into the linear motion of the work table is realized by the ball screw system with its transmission ratio i, which is defined as the distance

<sup>i</sup> <sup>¼</sup> <sup>h</sup>

Low-order modes are the main factors affecting the dynamic characteristics of the ball screw feed drive system of machine tools. Typically, the first axial and rotational modes of the ball screw show a dominant influence on the overall dynamics, while the relevance of higher-order

<sup>2</sup><sup>π</sup> (1)

parameter setting of the feed drive system [10].

40 New Trends in Industrial Automation

2.1. Lumped mass model of ball screw feed system

of travel h during one revolution of the shaft as the following:

modes for most technical applications is rather small [8].

system

In this model the inertial component parameters are defined as the following: rotary inertia of servomotor JM, screw shaft side equivalent rotary inertia JS, mass of base MB, screw shaft side equivalent mass MS, and mass of the work table MT.

The equivalent rigidity parameters in the model are defined as the following: equivalent torsional rigidity Krot, equivalent axial rigidity Kax, rigidity of ball screw nut Kn, and axial rigidity of the base KB.

The equivalent damping parameters are defined as the following: servomotor torsional damping CM, equivalent torsional damping Crot, screw shaft side damping CS, ball screw nut damping Cn, equivalent axial damping Cax, axial damping of the base CB, and axial damping of the guide Cg.

Figure 1. Typical structure of ball screw drive system. 1. Servomotor; 2. Coupling; 3. Fixed bearing; 4. Screw shaft; 5. Ball screw nut; 6. Work table; 7. Support bearing; 8. Machine bed; 9. Base.

Figure 2. Lumped mass model of ball screw feed system.

The DOF parameters of the lumped mass model are defined as the following: angular rotation of the servomotor θM, screw shaft angular rotation at the table position θS, axial displacement of the base XB, screw shaft axial displacement at the table position XS, and work table position XT. Therefore, the deformation of the equivalent springs is described as follows: equivalent torsional spring deformation θ<sup>M</sup> � θS, equivalent axial spring deformation XS � XB, axial spring deformation of the base XB, and screw nut contact deformation XT � XS � iθS.

The speed parameters of equivalent damping are defined as the following: servomotor equivalent damping speed θ\_ <sup>M</sup>, equivalent torsional vibration damping speed θ\_ <sup>M</sup> � <sup>θ</sup>\_ <sup>S</sup>, equivalent damping speed of the screw θ\_ S, equivalent damping speed of the base X\_ <sup>B</sup>, equivalent axial damping speed of screw <sup>X</sup>\_ <sup>S</sup> � <sup>X</sup>\_ <sup>B</sup>, equivalent damping speed of screw nut <sup>X</sup>\_ <sup>T</sup> � <sup>X</sup>\_ <sup>S</sup> � <sup>i</sup>θ\_ <sup>S</sup>, and speed of the work table X\_ <sup>T</sup>.

According to the Lagrange's equations of the second kind, the dynamic model of the ball screw feed drive system is built up. The total kinetic energy T, the potential energy U, and the dissipation function of the system can be expressed using equations (2) through (4):

$$T = \frac{1}{2} J\_M \dot{\theta}\_M^2 + \frac{1}{2} J\_S \dot{\theta}\_S^2 + \frac{1}{2} M\_B \dot{X}\_B^2 + \frac{1}{2} M\_S \dot{X}\_S^2 + \frac{1}{2} M\_T \dot{X}\_T^2 \tag{2}$$

where,

m ¼

JM 00 0 0 0 JS 000 0 0MB 0 0 00 0MS 0 00 0 0MT

c ¼

Figure 3. Simulation model of a ball screw feed drive.

k ¼

cM þ crot -crot 0 00 -crot crot-cS-i2cn 0 -icn icn

0 0cax þ cB -cax 0 0 icn -cax cax þ cn -cn 0 -icn 0 -cn cn þ cg

The dynamic model of the ball screw feed system shown in Eq. (8) was decomposed into three subsystems: screw shaft torsional vibration system, screw shaft axial vibration system, and the table vibration system. The simulation model of the ball screw feed system can be established as Figure 3. The input of the simulation model is the motor torque TM and the cutting force FC,

and the outputs are the table acceleration aT and displacement XT.

krot -krot 0 00 -krot krot-i2kn 0 -ikn ikn

Electromechanical Co-Simulation for Ball Screw Feed Drive System

43

0 0kax þ kB -kax 0 0 ikn -kax kax þ kn -kn 0 -ikn 0 -kn kn

http://dx.doi.org/10.5772/intechopen.80716

$$\mathcal{U} = \frac{1}{2}k\_{\text{rot}}(\theta\_{\text{M}} - \theta\_{\text{S}})^2 + \frac{1}{2}k\_{\text{B}}X\_{\text{B}}^2 + \frac{1}{2}k\_{\text{tr}}(X\_{\text{S}} - X\_{\text{B}})^2 + \frac{1}{2}k\_{\text{nut}}(X\_{\text{T}} - X\_{\text{S}} - i\theta\_{\text{S}})^2 \tag{3}$$

$$\begin{split} \mathbf{D} &= \frac{1}{2} \mathbf{C}\_{M} \dot{\boldsymbol{\theta}}\_{M}^{2} + \frac{1}{2} \mathbf{C}\_{nt} \left( \dot{\boldsymbol{\theta}}\_{M} - \dot{\boldsymbol{\theta}}\_{S} \right)^{2} + \frac{1}{2} \mathbf{C}\_{S} \dot{\boldsymbol{\theta}}\_{S}^{2} + \frac{1}{2} \mathbf{C}\_{B} \dot{\mathbf{X}}\_{B}^{2} \\ &+ \frac{1}{2} \mathbf{C}\_{\text{ax}} \left( \dot{\mathbf{X}}\_{S} - \dot{\mathbf{X}}\_{B} \right)^{2} + \frac{1}{2} \mathbf{C}\_{n} \left( \dot{\mathbf{X}}\_{T} - \dot{\mathbf{X}}\_{S} - i \dot{\boldsymbol{\theta}}\_{S} \right)^{2} + \frac{1}{2} \mathbf{C}\_{T} \dot{\mathbf{X}}\_{T}^{2} \end{split} \tag{4}$$

According to the definition of the system lumped mass, we have the independent coordinates system q as the following:

$$\mathbf{q} = \begin{pmatrix} \theta\_M & \theta\_S & X\_B & X\_S & X\_T \end{pmatrix}^T \tag{5}$$

The force inputs of the ball screw feed system are the servomotor torque TM and cutting force FC, and then the generalized forces Q of the system can be expressed as the following:

$$\mathbf{Q} = \begin{pmatrix} T\_M & \mathbf{0} & \mathbf{0} & \mathbf{0} & -F\_{\mathcal{C}} \end{pmatrix}^T \tag{6}$$

With L ¼ T � U, the Lagrangian function of the system about the generalized coordinate q and the generalized force Q can be calculated according to Eq. (7). Then, the matrix form of the lumped mass model of the ball screw feed system can be established as in Eq. (8):

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\mathbf{q}}}\right) - \frac{\partial L}{\partial \mathbf{q}} - \frac{\partial D}{\partial \dot{\mathbf{q}}} = \mathbf{Q} \tag{7}$$

$$\mathbf{m}\ddot{\mathbf{q}} + \mathbf{c}\dot{\mathbf{q}} + \mathbf{k}\mathbf{q} = \mathbf{Q} \tag{8}$$

where,

The DOF parameters of the lumped mass model are defined as the following: angular rotation of the servomotor θM, screw shaft angular rotation at the table position θS, axial displacement of the base XB, screw shaft axial displacement at the table position XS, and work table position XT. Therefore, the deformation of the equivalent springs is described as follows: equivalent torsional spring deformation θ<sup>M</sup> � θS, equivalent axial spring deformation XS � XB, axial spring deformation of the base XB, and screw nut contact deforma-

The speed parameters of equivalent damping are defined as the following: servomotor equiv-

According to the Lagrange's equations of the second kind, the dynamic model of the ball screw feed drive system is built up. The total kinetic energy T, the potential energy U, and the dissipation function of the system can be expressed using equations (2) thro-

kaxð Þ XS � XB

þ 1 2 CSθ\_ S 2 þ 1 2 CBX\_ <sup>B</sup> 2

Cn <sup>X</sup>\_ <sup>T</sup> � <sup>X</sup>\_ <sup>S</sup> � <sup>i</sup>θ\_

According to the definition of the system lumped mass, we have the independent coordinates

q ¼ ð Þ θ<sup>M</sup> θ<sup>S</sup> XB XS XT

The force inputs of the ball screw feed system are the servomotor torque TM and cutting force

Q ¼ ð Þ TM 00 0 �FC

With L ¼ T � U, the Lagrangian function of the system about the generalized coordinate q and the generalized force Q can be calculated according to Eq. (7). Then, the matrix form of the

> � ∂L <sup>∂</sup><sup>q</sup> � <sup>∂</sup><sup>D</sup>

FC, and then the generalized forces Q of the system can be expressed as the following:

lumped mass model of the ball screw feed system can be established as in Eq. (8):

∂L ∂q\_ 

d dt <sup>2</sup>

S

þ 1 2 CTX\_ <sup>T</sup>

damping speed of screw <sup>X</sup>\_ <sup>S</sup> � <sup>X</sup>\_ <sup>B</sup>, equivalent damping speed of screw nut <sup>X</sup>\_ <sup>T</sup> � <sup>X</sup>\_ <sup>S</sup> � <sup>i</sup>θ\_

<sup>M</sup> � <sup>θ</sup>\_ S <sup>2</sup>

> þ 1 2

<sup>M</sup>, equivalent torsional vibration damping speed θ\_

S, equivalent damping speed of the base X\_ <sup>B</sup>, equivalent axial

<sup>M</sup> � <sup>θ</sup>\_

<sup>T</sup> (2)

<sup>2</sup> (4)

<sup>T</sup> (5)

<sup>T</sup> (6)

<sup>∂</sup>q\_ <sup>¼</sup> <sup>Q</sup> (7)

mq€þcq\_þkq¼Q (8)

<sup>2</sup> (3)

knutð Þ XT � XS � iθ<sup>S</sup>

<sup>S</sup>, equivalent

<sup>S</sup>, and

tion XT � XS � iθS.

ugh (4):

alent damping speed θ\_

42 New Trends in Industrial Automation

damping speed of the screw θ\_

<sup>T</sup> <sup>¼</sup> <sup>1</sup> 2 JMθ\_ <sup>2</sup> <sup>M</sup> þ 1 2 JSθ\_ <sup>2</sup> <sup>S</sup> þ 1 2 MBX\_ <sup>2</sup> <sup>B</sup> þ 1 2 MSX\_ <sup>2</sup> <sup>S</sup> þ 1 2 MTX\_ <sup>2</sup>

krotð Þ θ<sup>M</sup> � θ<sup>S</sup>

<sup>D</sup> <sup>¼</sup> <sup>1</sup> 2 CMθ\_ M 2 þ 1 2 Crot θ\_

> þ 1 2

2 þ 1 2 kBX<sup>2</sup> <sup>B</sup> þ 1 2

Cax <sup>X</sup>\_ <sup>S</sup> � <sup>X</sup>\_ <sup>B</sup> <sup>2</sup>

speed of the work table X\_ <sup>T</sup>.

<sup>U</sup> <sup>¼</sup> <sup>1</sup> 2

system q as the following:

m ¼ JM 00 0 0 0 JS 000 0 0MB 0 0 00 0MS 0 00 0 0MT 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 k ¼ krot -krot 0 00 -krot krot-i2kn 0 -ikn ikn 0 0kax þ kB -kax 0 0 ikn -kax kax þ kn -kn 0 -ikn 0 -kn kn 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 c ¼ cM þ crot -crot 0 00 -crot crot-cS-i2cn 0 -icn icn 0 0cax þ cB -cax 0 0 icn -cax cax þ cn -cn 0 -icn 0 -cn cn þ cg 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5

The dynamic model of the ball screw feed system shown in Eq. (8) was decomposed into three subsystems: screw shaft torsional vibration system, screw shaft axial vibration system, and the table vibration system. The simulation model of the ball screw feed system can be established as Figure 3. The input of the simulation model is the motor torque TM and the cutting force FC, and the outputs are the table acceleration aT and displacement XT.

Figure 3. Simulation model of a ball screw feed drive.
